1. 03/01/15 1
Numerical Solution of Time-
Dependent Physical systems by
Means of Multi-dimensional Wave
Digital Filters
Jason Tseng
School of Engineering
University of Warwick, UK
2. 2
Outline
Physical systems modelling: Time-dependent PDEs
Distinct advantages of the MD-WDF modelling
MD-WDF modelling procedures
Continuous mapping: lumped electrical networks
Discrete mapping: bilinear transformation, wave digital filter
Examples and computational results
Sound wave propagation in a building (linear system).
Mindlin plate (thick plate) vibration (linear system).
Shallow water wave propagation (non-linear system).
Future work and conclusions.
3. 3
Time-dependent differential equation models
Original PDE models
Parabolic PDE:
Hyperbolic PDE:
Models can represent:
Finite-element spatial- and time- discretization of PDEs
Finite-difference spatial- and time- discretization of PDEs
Lumped electrical circuits with linear and/ or non-linear
capacitors and inductors.
( ) fauuc
t
u
d =+∇⋅∇−
∂
∂
( ) fauuc
t
u
d =+∇⋅∇−
∂
∂
2
2
on time.dependcanand,,,where dfac
4. 4
Approaches for numerical modelling
of time-dependent PDEs
Finite elements
Advantages:
Easy inclusion of local grid refinement
Easy handling of complex geometries
Disadvantages:
Computationally expensive
Hard to correctly set up the simulation plane
Finite differences
Advantages:
Computationally cheap
Easy to correctly set up the simulation plane
Disadvantages:
Difficulties in handling irregular boundaries
A Need for local grid refinement to increase the accuracy
5. 5
Approaches for numerical modelling
of time-dependent PDEs (cont.)
Multi-dimensional Wave digital Filters (MD-WDF)
A member of finite difference family:
Computationally cheap.
Easy to correctly set up the simulation plane.
Easy to handle complex geometries.
Conservation of passivity:
Achievement of full robustness due to positive port
resistances .
Guarantee to all numerical stabilities required of an accurate
numerical integration method.
6. 6
Advantages of the MD-WDF Model (cont.)
Fully local interconnectivity and massive parallelism
Behaviour of the equivalent passive dynamical discrete
system at any point in space is directly influenced only by
the points in its nearest neighbourhood.
Each point in the n-d grid can be updated simultaneously
when sufficient computing resources are available
High accuracy:
Low round-off noise characteristics of WDF structure
Suppression of parasitic oscillations of WDF structure
7. 7
MD-WDF modelling procedures
Multi-dimensional
Kirchhoff circuit
Discrete mapping
Multi-dimensional
Wave digital filters algorithm
System behaviour description
by lumped electrical network
Discrete passive
dynamic system description
Time-dependent
PDEs
Generalized
Trapezoidal rule
Multi-dimensional
Wave quantities
Kirchhoff’s current and voltage laws
Original passive
Physical system
MD DSP
MDKC
MD WDF
8. 8
Lumped electrical networks
Kirchhoff ‘s laws: n-port connection forming a loop.
Passive circuit elements of electrical networks.
Definition:
Schematic representation:
=
===
∑=
(voltages)0
(currents)
1
21
n
k k
n
u
iii
iRu 0=
≥=
≥
=
0)(),(
0),(
iLLiL
t
DL
Li
t
LD
u
=
−=
12
21
Riu
Riu
Resistor:
Inductor: Gyrator:
=
===
∑=
n
k k
n
i
uuu
1
21
0
Series connection Parallel connection
Ideal
transformer
−=
=
21
21
nii
unu
9. 9
Discrete mapping approach
Generalized trapezoidal rule (bilinear transformation) for
inductor:
Linear inductances:
Non-linear inductances:
0),,,,(where)),()(()( 4321 ≥=+±±= kzyxt LtzyxiDLDLDLDLu xxx
[ ]
delaytime:shift;spatial:,,
,,,,
2222
where))()(()()(
4321
tzyx
tzyx
zyxt
TTTT
TTTT
T
L
T
L
T
L
T
L
R
iiRuu
±±±=====
−−=−+
T
TxxTxx
0)(),)(())(())(())(()( 44332211 ≥=±±±= iLLiLDLiLDLiLDLiLDLu kkzyxt xxxxx
approximated
[ ]
4321,
2222
where)))(())(()()(
LLLLL
TTTT
R
iLiLR
L
u
L
u
zyxt
===≡===≡
−−=−+ TxxTxx
approximated
10. 10
MD wave quantities and adaptors.
Wave quantities:
Voltage waves (linear circuit elements):
Power waves (non-linear circuit elements):
Wave digital elements via bilinear transformation:
−
=
+
=
)power waveOutput(
2
)power wave(Input
2
R
Riu
b
R
Riu
a
−=
+=
wave)ltage(Output vo
wave)tage(Input vol
Riub
Riua
Resistor: Inductor: Gyrator:
)()( T−−= tatb
=
==
=
−=
es)(power wav
waves)(voltage
)()(
)()(
21
21
12
21
RRR
RRR
tatb
tatb
sourcevoltage:)(
0)(
)(2)(
te
tb
teta
=
=
Ideal
transformer
=
=
)(
1
)(
)()(
21
12
ta
n
tb
tnatb
11. 11
MD wave quantities and adaptors (cont.)
Relations of wave quantities in a n-port adaptor:
Voltage waves:
Series connection:
Parallel connection:
Power waves
Series connection
Parallel connection
∑
∑ =
=
=−=
n
j
jn
j j
k
kk nka
R
R
ab
1
1
,,1,
2
∑
∑ =
=
=+−=
n
j j
j
n
j
j
kk nk
R
a
R
ab
1
1
,,1,
1
2
nkaR
R
R
ab j
n
j
jn
j j
k
kk ,,1,
2
1
1
=−= ∑
∑ =
=
nk
R
a
R
R
ab
n
j j
j
n
j j
k
kk ,,1,
1
2
1
1
=+−= ∑
∑
=
=
12. 12
Stability conditions
Linear system.
CFL (courant-Friedrichs-levy) criterion to obtain the
maximum speed of wave propagation.
Least restriction on the density of the sampling in time for a
given density of sampling in space.
Non-linear system.
13. 13
Modelling example 1:
Sound wave propagation in a complex building
Governing equations of motion and continuity
System variables:
=
∂
∂
+
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
=
∂
∂
+
∂
∂
0),(),(),(),(
1
0),(),(
0),(),(
0),(),(
2
0
0
0
0
tv
z
tv
y
tv
x
tp
tc
tp
z
tv
t
tp
y
tv
t
tp
x
tv
t
zyx
z
y
x
xxxx
xx
xx
xx
ρ
ρ
ρ
ρ
soundofspeed:
airtheofdensity:
//,//,//withsvelocitiefluidacoustic:,,
pressureacoustic:
time:
,,scoordinatespaceofvector:
0
c
zvyvxvvvv
p
t
zyx
zyxzyx
−
−
−
−
−
−
ρ
x
15. 15
MDKC network description
Partial derivative operators:
Passivity of inductances
≤++<
≤<
≥−−−=
≥−=≥−=≥−=
2
0
2
0
0
2
0
2
0
000
0
,,0
0
0,0,0
c
r
c
r
L
LLL
zyx
zyx
zyxp
zzyyxx
ρ
δδδ
ρδδδ
δδδ
ρ
δρδρδρ
MDKC representation for 3D sound wave propagation in air
ztz
yty
xtx
DrDztD
DrDytD
DrDxtD
0
0
0
)(
)(
)(
−=±
±=±
±=±
δ
δ
δ
16. 16
Discrete mapping of MDKC
Generalized trapezoidal rule for inductors-shift operators:
MD voltage waves-port resistances:
sizesteptemporal:
,,insizesstepspatial:,,
where
],0,0,0[;
],,0,0[],,,0,0[
],0,,0[],,0,,0[
],0,0,[],,0,0,[
65
43
21
t
zyx
t
tztz
tyty
txtx
T
zyxTTT
T
TTTT
TTTT
TTTT
−
−
=
=−=
=−=
=−=
T
TT
TT
TT
≡′≡′≡′≡′
=≡=≡=≡
′=′=′=′=
======
t
p
t
z
t
y
t
x
t
z
zt
yx
yt
x
x
T
L
r
T
L
r
T
L
r
T
L
r
TT
r
r
TT
r
r
TT
r
r
rRrRrRrR
rRRrRRrRR
2
,
2
,
2
,
2
22
,
22
,
22
,,,,
,,
4321
0
3
0
2
0
1
414313212111
3107296185
δδδ
18. 18
Numerical results 1:
Sound wave propagation in 2D of complex building
Floor plan of one story building with location of sound sources
19. 19
Numerical results 2:
Acoustic pressure propagation in true 3D of 2 storeys
complex building
Floor plan of two storeys building with location of sound sources
20. 20
Modelling example 2:
Mindlin plate (thick plate) vibration
Governing equations of motion.
System variables:
=−
∂
∂
−
∂
∂
=−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
0
1
0
1
0
2
2
y
y
x
x
yx
w
y
v
t
Q
Gh
w
x
v
t
Q
Gh
y
Q
x
Q
t
v
h
κ
κ
ρ
=
∂
∂
−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
=
∂
∂
−
∂
∂
−
∂
∂
=+
∂
∂
−
∂
∂
−
∂
∂
=+
∂
∂
−
∂
∂
−
∂
∂
0
24
0
1
0
1
0
12
0
12
3
3
3
x
w
y
w
t
M
Gh
y
w
x
w
t
M
D
y
w
x
w
t
M
D
Q
y
M
x
M
t
wh
Q
y
M
x
M
t
wh
yxxy
yxy
yxx
x
yxyy
x
xyxx
ν
ν
ρ
ρ
shearinelasticityofmodulus:
)1(2
ncompressioandin tensionelasticityofdulusmodulus/mosYoung':
plateofrigidityflexural:
)1(12
ratiosPoisson'density,material,thicknessplate:),,(
platetheoflengthunitpermomentsbending:
platetheoflengthunitperforcesshearetransvers:),(
),(rotationsbendingtheofsvelocitie:),(
ntdisplacemeetransverstheofvelocity:
2
3
yx
ν
ν
νρ
ϕϕ
ϕϕ
+
=−
−
−
=−
−
−
∂
∂
=
∂
∂
=−
∂
∂
=−
E
G
E
Eh
D
h
),M,M-(M
QQ
t
w
t
w
t
w
v
xyyx
yx
y
y
x
x
0
12 2
2
2
2
2
2
2
23
2
=
∂
∂
+
∂
∂
−∇
∂
∂
−∇
t
w
hw
tGt
h
D ρ
κ
ρρ
+=
Sub-system 1
Sub-system 2
28. 28
Modelling example 3:
Non-linear water wave propagation
Governing equations of motion and continuity.
System variables:
( ) ( )
=
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
++
∂
∂
+
∂
∂
+
∂
∂
=
∂
∂
+−
∂
∂
+
∂
∂
+
∂
∂
0
0
0
21
1
2
2
2
1
2
2
1
2
1
1
1
t
h
hv
y
hv
x
y
gfv
y
v
v
x
v
v
t
v
x
gfv
y
v
v
x
v
v
t
v
η
η
.(constant)onacceleratigravity:
(constant)parameterCoriolis:
nt.displacemesurfacefree:),,(
.(constant)depthmean:
depth.total:
.y//,//withocitieswater vel:, 2121
g
f
tyxz
H
Hh
vxvvv
η
η
=
+=
29. 29
Graphical network description
Quantities normalization and equal physical dimension for
system variables:
Mesh equations representing MDKC.
parameterscaled:0
)(ithconstant w:0)(
where
ˆ,ˆ,ˆ,ˆ
3333
333
2
2
3
1
1
>−
=>=−
≡≡≡≡
ε
εεη
η
εε
ttttDv
v
h
h
vv
v
v
v
v
v
t
),,()ˆ,ˆ,ˆ( 32121 iiivv ≡η
=+++−−+
+−++
=++−−−−+
+−++
=−−−+++−
+−++
∑∑
∑ ∑
∑ ∑
∑ ∑
==
= = ++
= = ++
= =
0))(())((
)())(())((
0))(())((
)())(())((
0))(())((
)())(())((
13
2
1 313
2
1 3
2
1
2
1 3334334
232323231
2
1
2
1 2232232
131313132
2
1
2
1 11313
3
232
131
iittDiittD
iLDLittDittD
iittDiittDiR
iLDLittDittD
iittDiittDiR
iLDLittDittD
jjjj
j j tjjjj
g
j j vtvjjjj
g
j j vtvjjjj
ηη
31. 31
Discrete mapping of MDKC
Generalized trapezoidal rule for non-linear inductors-shift
operators:
MD power waves-port resistances:
Stability criterion:
[ ] [ ]
[ ] [ ] [ ]
rTvTT
T
TTTT
TTTT
tyx
t
tyty
txtx
ˆwhere
00
0,0
0,0
3
43
21
≡==
=
−==
−==
T
TT
TT
=
=
±
=
±
=
=
=
±
=
±
=
=
=
15,
ˆ
2
14;13,
ˆ
12;11,
ˆ
;
10,5,
ˆ
2
9,4;8,3,
ˆ
7,2;6,1,
ˆ
;
4,,1
ˆ
02
01
2
1
4
j
r
L
j
r
L
j
r
L
R
j
r
L
j
r
L
j
r
L
R
j
r
L
R
sj
v
v
v
sj
j
η
η
η
δ
δ
δ
δ
)0,,(min),0,,(maxwhere
)
3
2
(2,
)3/2g(H
max
)(2
)3/2g(H
1if
)3/2g(H
)(2
)3/2g(H
1if)
3
2
(2
),(
min
),(
max
max
min
max
3
2
min
max
min
max
3
2
min
max
max3
yxyx
Hg
H
v
HH
v
H
Hgv
yxyx
ηηηη
η
η
η
η
η
ε
η
η
η
η
εηε
==
+
+
+
≥
+
+
≤≠
+
+
≥
+
+
>≠+≥
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